| L(s) = 1 | + 5-s − 3·7-s + 3·11-s − 2·17-s + 6·19-s − 3·23-s − 4·25-s − 4·29-s + 6·31-s − 3·35-s − 7·37-s + 5·41-s + 6·43-s + 9·47-s + 2·49-s + 11·53-s + 3·55-s + 6·59-s − 8·61-s − 12·71-s − 15·73-s − 9·77-s − 12·79-s + 3·83-s − 2·85-s + 2·89-s + 6·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s + 0.904·11-s − 0.485·17-s + 1.37·19-s − 0.625·23-s − 4/5·25-s − 0.742·29-s + 1.07·31-s − 0.507·35-s − 1.15·37-s + 0.780·41-s + 0.914·43-s + 1.31·47-s + 2/7·49-s + 1.51·53-s + 0.404·55-s + 0.781·59-s − 1.02·61-s − 1.42·71-s − 1.75·73-s − 1.02·77-s − 1.35·79-s + 0.329·83-s − 0.216·85-s + 0.211·89-s + 0.615·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.022302279\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.022302279\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.06573850341560, −12.28995898429057, −12.12471690090684, −11.62037517187315, −11.17547691545938, −10.36122522893013, −10.16288253907439, −9.661551067073644, −9.147430452797118, −8.974531789324512, −8.307539204505503, −7.535608320241458, −7.231486999108331, −6.758227337665255, −6.133884173994037, −5.747809739059111, −5.497355242061129, −4.480975836262262, −4.133353006274726, −3.585784443368272, −2.993684154654587, −2.500708255565544, −1.782123225100904, −1.154989523000673, −0.4135106858461981,
0.4135106858461981, 1.154989523000673, 1.782123225100904, 2.500708255565544, 2.993684154654587, 3.585784443368272, 4.133353006274726, 4.480975836262262, 5.497355242061129, 5.747809739059111, 6.133884173994037, 6.758227337665255, 7.231486999108331, 7.535608320241458, 8.307539204505503, 8.974531789324512, 9.147430452797118, 9.661551067073644, 10.16288253907439, 10.36122522893013, 11.17547691545938, 11.62037517187315, 12.12471690090684, 12.28995898429057, 13.06573850341560
